Homo Actuarius Bayesianis

Bayesian fallacies are often the most trickiest.....

A classical example of a Bayesian fallacy is the so called "Prosecutor's fallacy" in case of DNA testing...

Multiple DNA testing (Source: Wikipedia)

A crime-scene DNA sample is compared against a database of 20,000 men.

A match is found, the corresponding man is accused and at his trial, it is testified that the probability that two DNA profiles match by chance is only 1 in 10,000.


Sounds logical, doesn't it?
Yes... 'Sounds'... As this does not mean the probability that the suspect is innocent is also 1 in 10,000. Since 20,000 men were tested, there were 20,000 opportunities to find a match by chance.

Even if none of the men in the database left the crime-scene DNA, a match by chance to an innocent is more likely than not. The chance of getting at least one match among the records is in this case:



So, this evidence alone is an uncompelling data dredging result. If the culprit was in the database then he and one or more other men would probably be matched; in either case, it would be a fallacy to ignore the number of records searched when weighing the evidence. "Cold hits" like this on DNA data-banks are now understood to require careful presentation as trial evidence.

In a similar (Dutch) case, an innocent nurse (Lucia de Berk) was at first wrongly accused (and convicted!) of murdering several of her patients.

Other Bayesian fallacies
Bayesian fallacies can come close to the actuarial profession and even be humorous, as the next two examples show:

1. Pension Fund Management
   
   
   It turns out that from all pension board members that were involved in a pension fund deficit, only 25% invested more than half in stocks.
   
   Therefore 75% of the pension fund board members with a pension fund deficit invested 50% or less in stocks.
   
   From this we may conclude that pension fund board members should have done en do better by investing more in stocks....
   
   
2. The Drunken Driver
   It turns out that of from all drivers involved in car crashes 41% were drunk and 59% sober.
   
   Therefore to limit the probability of a car crash it's better to drink...

It's often not easy to recognize the 'Bayesian Monster' in your models. If you doubt, always set up a 2 by 2 contingency table to check the conclusions....

Homo Actuarius
Let's  dive into the historical development of Asset Liability Management (ALM) to illustrate the different stages we as actuaries went through to finally cope with Bayesian stats. We do this by going (far) back to prehistoric actuarial times.
 

As we all know, the word actuary originated from the Latin word actuarius (the person who occupied this position kept the minutes at the sessions of the Senate in the Ancient Rome). This explains part of the name-giving of our species.

Going back further in time we recognize the following species of actuaries..

1. Homo Actuarius Apriorius
   This actuarial creature (we could hardly call him an actuary) establishes the probability of an hypothesis, no matter what data tell.
   
   ALM example: H₀: E(return)=4.0%. Contributions, liabilities and investments are all calculated at 4%. What the data tell is uninteresting.
   
   
2. Homo Actuarius Pragmaticus
   The more developed 'Homo Actuarius Pragamiticus' demonstrates he's only interested in the (results of the) data.
   ALM example: In my experiments I found x=4.0%, full stop.
   Therefore, let's calculate with this 4.0%.
   
   
3. Homo Actuarius Frequentistus
   In this stage, the 'Homo Actuarius Frequentistus' measures the probability of the data given a certain hypothesis.
   
   ALM example: If H₀: E(return)=4.0%, then the probability to get an observed value more different from the one I observed is given by an opportune expression. Don't ask myself if my observed value is near the true one, I can only tell you that if my observed value(s) is the true one, then the probability of observing data more extreme than mine is given by an opportune expression.
   In this stage the so called Monte Carlo Methods was developed...
   
   
4. Homo Actuarius Contemplatus
   The Homo Actuarius Contemplatus measures the probability of the data and of the hypothesis.
   
   ALM example:You decide to take over the (divided!) yearly advice of the 'Parameters Committee' to base your ALM on the maximum expected value for the return on fixed-income securities, which is at that moment  4.0%. Every year you measure the (deviation) of the real data as well and start contemplating on how the two might match...... (btw: they don't!)
   
   
5. Homo Actuarius Bayesianis
   The Homo Actuarius Bayesianis measures the probability of the hypothesis, given the data.  Was the  Frequentistus'  approach about 'modeling mechanisms' in the world, the Bayesian interpretations are more about 'modeling rational reasoning'.
   
   ALM example: Given the data of a certain period we test wetter the value of H₀: E(return)=4.0% is true : near 4.0% with a P% (P=99?) confidence level.

Knowledge: All probabilities are conditional
Knowledge is a strange  phenomenon...

When I was born I knew nothing about everything. When I grew up learned something about some thing. Now I've grown old I know everything about nothing. Joshua Maggid

The moment we become aware that ALL probabilities - even quantum probabilities - are in fact hidden conditional Bayesian probabilities, we (as actuaries) get enlightened (if you don't : don't worry, just fake it and read on)!

Simple Proof: P(A)=P(A|S), where S is the set of all possible outcomes.

From this moment on your probabilistic life will change.

To demonstrate this, examine the next simple example.

Tossing a coin

• When tossing a coin, we all know: P (heads)=0.5
• However, we implicitly assumed a 'fair coin', didn't we?
• So what we in fact stated was: P (heads|fair)=0.5
• 
  Now a small problem appears on the horizon: We all know a fair coin is hypothetical, it doesn't really exist in a real world as every 'real coin' has some physical properties and/or environmental circumstances that makes it more or less biased.
• We can not but conclude that the expression
  'P (heads|fair)=0.5'  is theoretical true, but has unfortunately no practical value.
• The only way out is to define fairness in a practical way is by stating something like:  0.4999≥P(heads|fair)≤0.5001
• Conclusion: Defining one point estimates in practice is practically  useless, always define estimate intervals (based on confidence levels).

From this beginners  example, let's move on to something more actuarial:

Estimating Interest Rates: A Multi Economic Approach

• Suppose you base your (ALM) Bond Returns (R) upon:
  μ= E(R)=4% and σ=2%
  
  
• Regardless what kind of brilliant interest- generating model (Monte Carlo or whatever) you developed, chances are your model is based upon several implicit assumptions like inflation or unemployment.
  
  The actual Return (R_t) on time (t) depends on many (correlated, mostly exogenous) variables like Inflation (I), Unemployment (U), GDP growth(G), Country (C) and last but not least  (R_[t-x]).
  
  A well defined Asset Liability Model should therefore define (R_t) more on basis of a 'Multi Economic Approach'  (MEA) in a form that looks more or less something like: R_t = F(I,U,G,σ,R_[t-1],R_[t-2],etc.)
  
  
• In discussing with the board which economic future scenarios will be most likely and can be used as strategic scenarios, we (actuaries) will be better able to advice with the help of MEA. This approach, based on new technical economic models and intensive discussions with the board, will guarantee  more realistic output and better underpinned decision taking.

Sources and related links:
I. Stats....
- Make your own car crash query
- Alcohol-Impaired Driving Fatalities (National Statistics)
- D r u n k D r i v i n g Fatalities in America (2009)
- Drunk Driving Facts (2006)

II. Humor, Cartoons, Inspiration...
- Jesse van Muylwijck Cartoons (The Judge)
- PHDCOMICS
- Interference : Evolution inspired by Mike West

III. Bayesian Math....
- New Conceptual Approach of the Interpretation of Clinical Tests (2004)
- The Bayesian logic of frequency-based conjunction fallacies (pdf,2011)
- The Bayesian Fallacy: Distinguishing Four Kinds of Beliefs (2008)
- Resource Material for Promoting the Bayesian View of Everything
- A Constructivist View of the Statistical Quantification of Evidence
- Conditional Probability and Conditional Expectation
- Getting fair results from a biased coin
- INTRODUCTION TO MATHEMATICAL FINANCE

Humor: Scrambled Actuarial Reporting

Some actuaries are convinced that adding more important details really helps. With more details and more information you are able to explain you models better and as we all know: better communication is key in actuarial science.


Here is an example of detailed information (click on the image!)

Some(times) details don't matter
Unfortunately more information and more details generally disturb efficient decision making. The next text shows that some details don't really matter.

Smoe acaruites are covcnined taht adding mroe imnrpotat deaitls rlaely hleps. Wtih more dleitas you are albe to eplaxin you mlodes bteter and as we all konw: btteer cmniutcoiaomn is key in aratiuacl sieccne.

Sirnpigrulsy tihs is not ture. Tihs txet sowhs taht smoe daeilts dno't rlaley mttear.

The arutacial aidnceue isn't rlaley istretneed in the daeilts, but in caelr ipunt (fsrit ltteer of a wrod) and oumotces (last letetr of a word). The dtilaes (letetrs) in bweteen can be mexid up in evrey rodnam oerdr you lkie. Keep in mnid tihs iponmatrt lsosen in your nxet peeiatntsorn.

Explanation
According to a study at Cambridge University, to read and understand a text well, it doesn't matter in what order the letters in a word are placed. The only condition is that the first and last letter of each word remain the same. The rest can be a total mess up. This is because the human mind does not read every letter by itself but the word as a whole.

DIY
Let's conclude with an 'example text' for the opening-slide of you next board presentation:

Daer Board mrebmes,

Agtlhouh we hvae to tkae fetdanmaunl dniecioss tdoay, it wlil not be ncseresay to udasnertnd or dcssius all knid of tcihcenal dtileas.

The relust of my avicde is pertseend in scuh a way as to esurne taht we can stcik to the mian ptinos and hneieadls.

The vrey fcat that you wree albe
to raed and udnreastnd tihs txet,
greauetans taht we wlil hvae a
sefscuucsl mtineeg.

Yuor aivdosr

Scramble your own opening-slide text for your next presentation at:

Words Scrambler

No doubt, your next report will be actuarial scrambled.... ;-)

Related sources and links
- Words Scrambler
- MRC Cognition and Brain Sciences Unit
- All My Faves

The Ten Actuarial Commandments

We all (think to) know The Ten Commandments from the holy scripts by heart, do we?

Now close your eyes to see how far you can get in quoting those simple ten guidelines in life.............

The Ten Commandments for Investors
Just like the Ten Commandments for Man, God - more specific - created The Ten Commandments for Investors. Let's compare the two, while - at the same time - you can check out your Commandment-Memory on Man as well:

Risk-Return-Supervision Development
As you may have noticed, The Ten Commandments are a mix of rules-based and principles-based principles.

Just as in our own life, it's interesting to see how we apply and implement these two different kind of rules during the evolution of a financial institution (insurance company, pension fund, bank, etc.):

In time, the ideal supervision model consists of three phases:

• Phase I: No rules
  In this phase we cannot value or the company. Chances are substantial the company is 'at risk'.
  
  
• Phase II: Rules-Based Supervision
  In phase Ia 'Rules' are mostly perceived as 'Have to's" . As a result Risk will be reduced, but Return as well. Once the board, actuaries and financial specialists are becoming aware and will see the advantages and new possibilities of managing risk. 'Have to's" will develop into 'Want to's" . The Risk-Return Ratio will increase  and even a better Return will result.
  
  
• Phase III: Principles-Based Supervision
  Just like with the implementation of Rules-based Supervision, in case of Principles-Based Supervision, the Financial Institution needs time to adept to the new situation. At first there might be a unbalance between Risk and Return. It takes time to calibrate Risk and Return again.
  
  After a while actuaries, investors and management will translate Rules-Based principles into own rules that fits the company's specific risk in an optimal way. The company will be able to take more risk and to optimize its own Risk-Return Ratio.

Take a look at your own company's development and see for yourself where you fit in on the Risk-Return-Supervision lines....

It might be possible that you have to conclude that you aren't able to increase your Risk-Return ratio in the end. In this case it's likely you've become (so called) 'Supervisory Compliant': Your risk appetite probably corresponds more or less with the supervisor's minimal risk view. Instead of redefining your own risk appetite and restructuring your products from a risk-management perspective you merely implied new regulations and supervisor guidelines. As a result your Return and Risk-Return Ratio implode....

Ten Actuarial Commandments
Having learned the possible effects of supervisory rules in practice, we may now conclude with The Ten Commandments for Actuaries.

The Ten Commandments for Actuaries

1. There's only one God, as he's omnipotent he's also an actuary.
   As you're only an actuary: be humble.....    Remember: As God wants something in Return, you'll have to take Risk!!
2. Reality can't be comprised in a model.
   Use your brains. A model is a help, not a decision machine. Don't mix up God with Risk or Chaos. Chaos for us humans (actuaries) can be defined as "Unrecognized Order" (quote). 
3. Never blame anything or anyone than yourself for an unexpected or negative outcome.
4. Be consistent, act sustainable. But change your opinion just in time, if circumstances or facts urge you to do so.
5. Alway show respect to others, even if you think different. Appreciate where you come from. Nobody is perfect, not even you.
6. As there is no 'right' model, never criticize other models, actuaries or other people. Try to give your opinion without slaughtering the other.
7. Never advice or state anything you do not really mean or cannot defend.If you're not sure or don't know, tell it or get help.
8. Always cite your sources or give credits to others that helped you.
9. Don't 'steal' the advice.
   Never include the final decision to be taken in your advice. Wrap up arguments, consequences and present scenario's so the board has to make a choice and not you.
10. Don't get carried away by results, reports or performances of others.
    Stick to your own consistent approach.

Apply supervisory rules and actuarial commandments in a conscious way...